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Structural optimization of radial containment ultrasonic suspension bearing
Micro/Nano Technology and Fine Mechanics | Updated：2021-11-01
|
Structural optimization of radial containment ultrasonic suspension bearing
- Optics and Precision Engineering
- Affiliations：
1.School of Mechanical and Electrical Engineering， Dalian Nationalities University， Dalian 116650， China
2.School of Mechanical Engineering， Dalian Jiaotong University， Dalian 116028， China
- Author bio：
- Funds：
- DOI：10.37188/OPE.20212910.2375 CLC： TN384
Published：15 October 2021，
Received：23 June 2021，
Revised：10 August 2021，
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Cite this article
李东明,王万雷,贾颖等.径向包容式超声悬浮轴承结构优化[J].光学精密工程,2021,29(10):2375-2385.
LI Dong-ming,WANG Wan-lei,JIA Ying,et al.Structural optimization of radial containment ultrasonic suspension bearing[J].Optics and Precision Engineering,2021,29(10):2375-2385.
李东明,王万雷,贾颖等.径向包容式超声悬浮轴承结构优化[J].光学精密工程,2021,29(10):2375-2385. DOI： 10.37188/OPE.20212910.2375.
LI Dong-ming,WANG Wan-lei,JIA Ying,et al.Structural optimization of radial containment ultrasonic suspension bearing[J].Optics and Precision Engineering,2021,29(10):2375-2385. DOI： 10.37188/OPE.20212910.2375.
Structural optimization of radial containment ultrasonic suspension bearing
径向包容式超声悬浮轴承结构优化 Structural optimization of radial containment ultrasonic suspension bearing 1 first-author 李东明（1971-），男，辽宁大连人，博士，教授，2003年于大连理工大学获得博士学位，主要从事传感器与执行器及精密加工方面的研究。E-mail： lidong-ming2001@163. com李东明（1971-），男，辽宁大连人，博士，教授，2003年于大连理工大学获得博士学位，主要从事传感器与执行器及精密加工方面的研究。E-mail： lidong-ming2001@163. com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871825&type= 22.01332855 32.00401306 李东明 （1971-），男，辽宁大连人，博士，教授，2003年于大连理工大学获得博士学位，主要从事传感器与执行器及精密加工方面的研究。E-mail： lidong-ming2001@163. com 李东明 （1971-），男，辽宁大连人，博士，教授，2003年于大连理工大学获得博士学位，主要从事传感器与执行器及精密加工方面的研究。E-mail： lidong-ming2001@163. com lidong-ming2001@163. com 1 corresp E-mail： wwl@dlnu.edu.cn E-mail： wwl@dlnu.edu.cn 王万雷（1973-），男，河北张家口人，博士，副教授，2006年于大连理工大学获得博士学位，主要从事从事智能制造，工业工程方面的研究。E-mail：wwl@dlnu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871843&type= 22.01332855 32.00399780 王万雷 （1973-），男，河北张家口人，博士，副教授，2006年于大连理工大学获得博士学位，主要从事从事智能制造，工业工程方面的研究。E-mail： wwl@dlnu.edu.cn wwl@dlnu.edu.cn 2 2 大连民族大学 机电工程学院，辽宁 大连 116650 大连民族大学 机电工程学院，辽宁 大连 116650 School of Mechanical and Electrical Engineering， Dalian Nationalities University， Dalian 116650， China School of Mechanical and Electrical Engineering， Dalian Nationalities University， Dalian 116650， China 大连交通大学 机械工程学院，辽宁 大连 116028 大连交通大学 机械工程学院，辽宁 大连 116028 School of Mechanical Engineering， Dalian Jiaotong University， Dalian 116028， China School of Mechanical Engineering， Dalian Jiaotong University， Dalian 116028， China 为了提高超声悬浮轴承性能，提出了一种径向包容式超声悬浮轴承结构优化设计方法。基于扁壳理论和弹性力学建立了轴承壳体结构固有频率和径向挠度的数学模型，以壳体结构的长、半径、壁厚3个参数为设计变量，以壳体结构的固有频率和径向挠度作为优化目标函数，采用多目标遗传算法进行求解，从求解得到的Pareto解集中选择一组最符合设计要求的解，即长40 mm，半径10 mm，壁厚2 mm作为优化结果。为了验证本文优化方法的有效性，制作了3组轴承进行对比实验。实验结果表明：优化得到的轴承结构兼顾了超声悬浮轴承对固有频率和径向挠度的性能要求，可以产生较大的径向悬浮力，本文的研究方法和结果对设计和制造新型超声悬浮轴承具有重要的指导意义。 To improve ultrasonic suspension bearing performance， a design method for structural optimization of radial inclusive ultrasonic suspension bearings is proposed. A mathematical model of natural frequency and radial deflection of bearing shell structure was established based on flat shell and elasticity theory. The three parameters shell structure， length， and radius and wall thickness， were taken as the design variables， and the natural frequency and radial deflection of shell structure were taken as the functional optimization objective. A multi-objective genetic algorithm was used to solve this problem. From the Pareto solution set obtained， a set of solutions most in line with design requirements （40 mm in length， 10 mm in radius， and 2 mm in wall thickness） were selected as the optimization results. To verify the efficacy of the optimization method in this paper， three groups of bearings were made for comparison testing. Experimental results show that the optimized bearing structure takes ultrasonic suspension bearing natural frequency and radial deflection performance requirements into account， and can produce a large radial suspension force. This research method and the results of this study can provide important and significance guidance for the design and manufacture of new ultrasonic suspension bearings. 超声悬浮轴承 多目标遗传算法 优化设计 气体挤压膜 ultrasonic suspension bearing multi-objective genetic algorithm optimized design gas squeeze film 1 引 言 1 随着精密加工制造技术的迅速发展，传统的接触式轴承因摩擦与磨损大、寿命低及转速低等缺点，已无法满足半导体、医疗器械等行业对加工精度的要求。超声悬浮轴承利用振动获得承载能力，其结构简单、体积小，不需要额外的压力源，在转子启停阶段也能保持承载能力，具有良好的发展前景。 超声悬浮轴承主要由压电控制电源、换能器和悬浮轴组成，根据功率大小可划分为两类。第一类是采用压电叠堆式换能器的超声悬浮轴承，使用超声变幅杆将压电叠堆产生的机械振动位移放大 ［ 1 1 ］ 。此类轴承换能器中的压电振子沿厚度方向伸缩振动并采用多层堆叠的方式结合在一起，其特点是功率大、输出振幅大、悬浮质量大，但体积大，发热量也较大。第二类超声悬浮轴承使用片状压电材料来制作换能器，压电振子粘贴在基体结构上并沿长度方向伸缩振动，通过耦合基体放大机构来提高作动性能，同时还可利用压电材料的频率性能来提高应变输出，使得结构在某一方向发生快速振动。其特点是结构紧凑、能耗低，易于控制，适合小型微型应用场景。 目前，对超声悬浮轴承中压电叠堆式换能器变幅杆的设计、优化和仿真分析的研究已较为成熟 ［ 2 2 - 3 3 2-3 ］ 。对于第二类超声悬浮轴承，由于功率较小，需要通过合适的基体机构来放大压电片的伸缩振动，因此换能器的基体结构多种多样，其中应用最多的为径向包容式结构和铰链结构。1991年，东京理工大学的Yoshimoto等 ［ 4 4 ］ 提出了一种将压电片嵌入轴承表面的系统。该轴承由滑块、压电驱动器和配重组成。当交变电压作用于压电驱动器时，滑块轴承在配重的作用下开始振动，并在轴承和导轨表面之间产生挤压膜。1997年，Yoshimoto等人 ［ 5 5 - 6 6 5-6 ］ 又提出了一种使用弹性铰链减小轴承尺寸的挤压膜空气轴承，并将这种轴承应用于方形线性导轨。但轴承运行在几千赫的振动频率时会产生噪音。2010年，Stolarski等人设计了3种结构的压电换能器 ［ 7 7 ］ ，它们均使用压电片作为驱动源，并通过有限元软件来仿真计算轴承几何形状的共振频率，实验表明轴承壳固定在壳体上的方式和轴承壳材料的机械性能有关。2016年，Stolarski又提出了一种利用压电片将滑动圆轴承间隙变形为三叶轴承间隙的弹性铰链轴承 ［ 8 8 ］ 。实验和数值计算结果表明，该轴承能够产生挤压膜压力效应。2018年，湖南大学Feng等 ［ 9 9 ］ 提出并设计一种新型的超声波气体轴承，该轴承使用贴片式压电陶瓷驱动可倾瓦轴承高频振动从而产生悬浮效果。2019年，东北大学姚红良等设计了一种超声挤压轴承 ［ 10 10 ］ ，并将它应用在小型或微型转子系统上，对该系统进行动力学特性分析并通过试验研究证明了理论分析的正确性。 以上对国内外研究现状的分析表明，目前第二类超声悬浮轴承的基体结构设计多依靠经验和仿真试验来确定形状和结构尺寸，缺乏理论支撑和通用性强的结构优化设计方法。本文设计了一种径向包容式超声悬浮轴承，并以轴承结构为对象，提出了基于多目标遗传算法的壳体结构优化方法，最后通过试验测试对比分析验证了该方法的正确性。 2 超声悬浮轴承结构与工作原理 1 如 图1 图1 所示，轴承主要由矩形压电陶瓷片和包容式金属圆柱筒组成，圆筒外侧间隔120°均布有轴向肋板。压电片通过丙烯酸结构胶粘贴在圆柱上。 当对压电片通入高频激振信号时，压电片带动金属圆筒产生周期性弹性变形。当压电片的驱动频率等于金属圆筒的固有频率时，圆柱筒内壁结构产生谐振变形，径向位移达到最大值，此时气膜的挤压角频率就等于固有频率。 在金属圆筒高频挤压作用下，轴与套筒微小间隙中的空气会形成具有稳定厚度和一定承载力的挤压气膜，忽略挤压气膜在轴承边缘存在的微小能量泄漏。在黏性流体动力学中，由N-S方程和连续方程可推导出直角坐标系下气体挤压膜压力分布的三维Reynolds方程： <math display="block" id="M1"><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mi>ρ</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">12</mn><mi>μ</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>p</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mi>ρ</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">12</mn><mi>μ</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>p</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>p</mi><mi>h</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871872&type= 55.79533768 9.39799976 . 对方程进行无量纲化得无量纲形式Reynolds方程： <math display="block" id="M2"><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>θ</mi></mrow></mfrac><mfenced separators="|"><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mfrac><mrow><mo>∂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo>∂</mo><mi>θ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac><mfenced separators="|"><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mfrac><mrow><mo>∂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2</mn><mi>σ</mi><mfrac><mrow><mo>∂</mo><mi>P</mi><mi>H</mi></mrow><mrow><mo>∂</mo><mi>τ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871880&type= 61.21400070 9.31333351 ， 式中：无量纲参量为 Z = z/L ， P = p/p 0 ， T = wt ， p 0 为周围环境压力； r 为轴承金属圆柱筒内半径； L 为轴承长； ω 为挤压角频率； μ 为气体的动力黏度； H = h/h 0 ， h 0 为轴承的初始间隙； h 为气膜的膜厚函数， <math id="M3"><mi>σ</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>μ</mi><mi>ω</mi><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi></mrow><mrow><msubsup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871890&type= 19.47333336 8.89000034 为挤压数。 使用有限差分方法求解 式（2） 式（2） 得到挤压气膜压力分布函数 P （ θ ， z ） ，将它在求解域上进行积分即可得到瞬时挤压气膜承载力： <math display="block" id="M4"><mi>F</mi><mo>=</mo><mo>∬</mo><mi>P</mi><mi mathvariant="normal">d</mi><mi>θ</mi><mi mathvariant="normal">d</mi><mi>Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871900&type= 21.08200073 6.68866634 . 由气膜承载力的推导过程进一步分析可知：在其他参数不变的情况下，一定范围内增大振幅和挤压角频率均可增大超声悬浮轴承瞬时挤压气膜承载力 F 。 2.1 轴承结构优化目标的数学模型 2 超声悬浮轴承的承载能力是一项重要的性能指标，其大小决定了轴承的实际使用场合和应用价值。为此，超声悬浮轴承的主要设计目标是获取最大的承载能力。影响轴承承载能力的参数主要是激励振幅和激振频率。 获取更大承载能力可以从两个方面着手，一方面是增大激励振幅，压电材料的形变程度只与驱动电压有关。因此，只有通过改变轴承壳体结构的局部刚度情况以获取更大的变形能力，即轴承与轴配合表面的位移变化值（即挠度）可设定为目标函数。另一方面，当激振频率等于轴承壳体的固有频率时产生共振，此时振幅较大，因此激振频率的选择应当使频率正好工作在结构的固有频率处。由雷诺方程的挤压数表达式可知激振频率与挤压数成正比，频率越大挤压效果越明显，轴承的承载能力越强。除此之外，为了防止轴承在气体挤压膜边缘处发生压力泄漏现象 ［ 11 11 ］ ，应尽量选取较大的激振频率。所以，轴承壳体的固有频率可设定为承载力优化的第二个目标函数。综上，本文选择轴承圆筒壳体的固有频率和挠度作为优化目标并建立其数学模型。 2 . 1 . 1 固有频率函数 3 在研究轴承结构的固有频率时可将轴承圆筒简化为圆柱壳体。建立的柱坐标系如 图2 图2 所示，以 u ， v ， w 分别表示3个坐标方向的位移分量，图中 L 为圆柱壳体的长度， a 为圆柱壳体的平均半径， h 为圆柱壳体的厚度。 根据扁壳理论或唐奈运动微分方程可得： <math display="block" id="M5"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>v</mi></mrow><mrow><mo>∂</mo><mi>z</mi><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>μ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>ω</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>v</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo>∂</mo><mi>z</mi><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>ω</mi></mrow><mrow><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>v</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>μ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>ω</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">12</mn></mrow></mfrac><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871926&type= 107.27265930 30.98800087 ， 式中： E 为壳体材料的弹性模量； u 为壳体材料的泊松比； ρ 为壳体材料的密度； g 为重力加速度； t 为时间。 <math display="block" id="M6"><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871936&type= 58.67399979 9.22866726 . 在超声悬浮轴承的壳体结构中主要关心 w 方向的振动，因此可以忽略壳体中面切线方向（ u，v ）的惯性，根据无力矩理论，略去 式（4） 式（4） 中的力矩项。对 式（4） 式（4） 进行微分运算，相互代换后得： <math display="block" id="M7"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mi>μ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><mi>z</mi><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>ρ</mi></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871946&type= 65.70133209 30.47999954 . 根据纯力矩理论，不考虑薄膜抗力作用，将 式（4） 式（4） 中内力矩项和惯性力项组成振动方程： <math display="block" id="M8"><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>ρ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mi>E</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ω</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871957&type= 52.15466690 8.89000034 . 经过以上简化，在唐纳尔形式的运动微分方程基础上将一般振动情况分别按无力矩理论（薄膜理论）和纯力矩理论（薄板弯曲理论）分别计算，最后综合两者的结果，可以得到圆柱壳体类结构精度较高的固有频率解 ［ 12 12 ］ 。 对 式（4） 式（4） 按无力矩理论进行化简计算，首先设其解的形式为： <math display="block" id="M9"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>n</mi><mi>θ</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>t</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>n</mi><mi>θ</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>t</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>n</mi><mi>θ</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>t</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871968&type= 40.97866440 15.74800014 ， 式中： U 1 ， V 1 ， W 1 为振型函数， n 为周向波数， ω 1 为圆频率。将 式（7） 式（7） 代入 式（5） 式（5） 得到： <math display="block" id="M10"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mi>μ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>n</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><msub><mrow/><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871979&type= 64.51599884 30.14133072 ， 其中： <math display="block" id="M11"><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871988&type= 94.40332794 9.31333351 . 进一步整理可得： <math display="block" id="M12"><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21871999&type= 46.22800064 8.04333401 ， <math display="block" id="M13"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872011&type= 25.99266624 18.37266731 ， <math display="block" id="M14"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872019&type= 25.99266624 18.37266731 . 设 <math id="M15"><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872031&type= 10.66800022 4.48733330 的解为 <math id="M16"><msup><mrow><mi>e</mi></mrow><mrow><mi>s</mi><mi>x</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872040&type= 3.38666677 3.04800010 形式，则有 <math id="M17"><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872052&type= 27.68600082 4.06400013 可得： <math id="M18"><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mo>-</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>±</mo><mroot><mrow><msubsup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872061&type= 36.40666580 9.05933285 。 此时应该对低阶固有频率和高阶固有频率分别进行考虑。 当 <math id="M19"><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>≪</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872071&type= 16.93333435 8.63599968 时，取 <math id="M20"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≅</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872082&type= 19.89666748 8.63599968 ， <math id="M21"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≅</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872095&type= 19.38866615 8.63599968 ，得 <math id="M22"><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>±</mo><mroot><mrow><mfrac><mrow><mi>g</mi><mi>E</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872103&type= 43.77266693 10.75266743 。 当 <math id="M23"><mfrac><mrow><mi>g</mi><mi>E</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>≫</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872113&type= 16.93333435 8.63599968 ，有 <math id="M24"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><mi>g</mi><mi>E</mi></mrow><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow><mrow/></mroot><mo>≫</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872124&type= 20.57399940 9.90600014 ，得： <math display="block" id="M25"><msup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>±</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><mi>g</mi><mi>E</mi></mrow><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mo>≅</mo><mo>±</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>a</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><mi>ρ</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>g</mi><mi>E</mi></mrow></mfrac></mrow><mrow/></mroot><mo>≅</mo><mo>±</mo><msup><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872136&type= 76.28466797 10.75266743 . 可求得振型函数 W 1 ： <math display="block" id="M26"><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872146&type= 88.90000153 4.48733330 . 其中 c 1 ， c 2 ， c 3 ， c 4 为待定常数。根据 式（8） 式（8） 可求出： <math display="block" id="M27"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>z</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872157&type= 94.91133118 9.99066734 ， 其中： <math display="block" id="M28"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>a</mi></mrow></mfrac><mfenced separators="|"><mrow><mi>μ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">2</mn><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow></mfrac><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872167&type= 47.07466888 19.81200027 . 将两端自由支承边界条件： <math id="M29"><mi>z</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872177&type= 8.38199997 2.79399991 及 <math id="M30"><mi>z</mi><mo>=</mo><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872189&type= 7.70466709 2.79399991 处， <math id="M31"><mi>v</mi><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872198&type= 18.79599953 7.53533268 代入 式（10） 式（10） ，得到特征方程为： <math display="block" id="M32"><mn mathvariant="normal">2</mn><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>μ</mi><msubsup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>l</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872208&type= 61.29866791 9.31333351 . 当 <math id="M33"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872219&type= 16.17133331 3.72533321 或 <math id="M34"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>l</mi><mo>=</mo><mi>m</mi><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>m</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872231&type= 39.96266937 4.48733330 时可得： <math display="block" id="M35"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi></mrow><mrow><mi>n</mi><mi>l</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mroot><mrow><mfrac><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>E</mi></mrow><mrow><mi>ρ</mi><mo>/</mo><mi>g</mi></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872241&type= 29.97200012 10.15999985 . 对 式（1） 式（1） 按照纯力矩理论计算，设其解的形式为： <math id="M36"><mi>w</mi><mo>=</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>n</mi><mi>θ</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872251&type= 38.43866730 4.48733330 代入 式（6） 式（6） 得： <math display="block" id="M37"><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo>∇</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872261&type= 32.42733002 4.23333359 ， <math display="block" id="M38"><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>ρ</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>g</mi><mi>E</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872270&type= 31.66533470 9.73666668 . 于是可得： <math display="block" id="M39"><mtable><mtr><mtd><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>z</mi><mo>+</mo></mtd></mtr><mtr><mtd><mi>B</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>z</mi><mo>+</mo><mi>C</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>z</mi><mo>+</mo></mtd></mtr><mtr><mtd><mi>D</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>z</mi><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872280&type= 68.83399963 37.25333405 式中 A，B，C，D 为任意常数。同样引入两端自由支承边界条件： <math id="M40"><mi>z</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872291&type= 8.21266651 2.79399991 及 <math id="M41"><mi>z</mi><mo>=</mo><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872303&type= 7.53533268 2.79399991 处， <math id="M42"><mi>w</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi></mrow><mrow><mo>∂</mo><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872313&type= 24.97666740 8.04333401 。 将边界条件代入 式（14） 式（14） ，得： <math display="block" id="M43"><mn mathvariant="normal">2</mn><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>l</mi><mo>×</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mfenced separators="|"><mrow><mroot><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow></mfenced><mi>l</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872325&type= 68.15666962 11.26066685 . 求得： <math display="block" id="M44"><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mfrac><mrow><mi>E</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>g</mi></mrow><mrow><mn mathvariant="normal">12</mn><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mi>ρ</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872337&type= 51.30799866 10.83733368 . 综上所述，最后结合无矩理论和纯力矩理论的结果可得： <math display="block" id="M45"><mi>ω</mi><mo>=</mo><mroot><mrow><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872348&type= 22.86000061 4.74133301 . 由此求出壳休的固有频率： <math display="block" id="M46"><mi>f</mi><mo>=</mo><mfrac><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872357&type= 11.76866722 7.61999989 . 2 . 1 . 2 挠度函数 3 轴承圆筒外侧的肋板将圆筒壳体分隔成3个对称的振动区域。由肋板隔开的区域大小相对于整个轴承圆筒的曲率半径而言较小，可将每个振动区域的圆筒壁展开，简化为粘贴有压电陶瓷片的弹性薄板，如 图3 图3 所示。图中 l 为薄板的长边， β 为短边，压电片的极化方向与 z 方向平行。 在电压作用下，压电材料延长度方向产生周期性的伸缩变形，压电材料的应变又通过理想黏结层传递给薄板，最终使薄板基体结构产生变形。 由压电材料的性质可知压电片内产生的驱动应变为： <math display="block" id="M47"><mi>Λ</mi><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub><mo>/</mo><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872372&type= 20.99733353 4.14866638 ， 其中： d 31 为压电陶瓷的压电常数， t c 为压电片厚度， V （ t ）代表驱动电压。 以压电片和压电片与薄板接触部分组成的组合梁（ 图4 图4 ）整体为研究对象，分析时做如下假设： （1）压电片与薄板基体理想粘贴； （2）组合梁发生纯弯曲变形； （3）假设组合梁横截面上的纵向应变沿横截面高度按直线规律变化。 根据组合梁理论可知，截面的中性轴与截面对称轴之间的距离为： <math display="block" id="M48"><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi><mi>o</mi></mrow></msub></mrow><mrow><mo>∑</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872386&type= 22.09799957 10.58333302 ， 式中： E i 为组合梁各组成部分材料的弹性模量， A 为组合梁各组成部分的面积， S io 为组合梁各组成部分的面积相对于组合梁横截面对称轴的面积。 可得到组合梁横截面沿 x 方向的应变分布函数： <math display="block" id="M49"><mi>ε</mi><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow><mrow><mi>H</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>-</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow></mfrac><mi>Λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872392&type= 25.31533241 8.72066593 ， 式中 H 为横梁厚度。压电片产生的内力弯矩 M ，可通过如下积分求得： <math display="block" id="M50"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>a</mi></mrow></msub><msubsup><mo>∫</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow><mrow><mi>H</mi><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow><mi>ε</mi><mi>z</mi><mi mathvariant="normal">d</mi><mi>z</mi><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>a</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mo>-</mo><mi>H</mi><mo>/</mo><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mi>ε</mi><mi>z</mi><mi mathvariant="normal">d</mi><mi>z</mi><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mfrac><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>a</mi></mrow></msub><msub><mrow><mi>ε</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>H</mi><mo>/</mo><mn mathvariant="normal">2</mn><mo>-</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow></mfrac><mfenced open="{" close="" separators="|"><mrow><mfrac><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">24</mn></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced><mo>+</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced open="" close="}" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>z</mi></mrow><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><msubsup><mrow><mi>z</mi></mrow><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872398&type= 60.87533569 31.07266808 式中： z a = H /2 -t c ， E a ，E b 分别为压电片材料和薄板的弹性模量， l a 为组合梁的长度。 为了简化模型，在研究弹性薄板弯曲变形时压电片对基体结构施加一对弯矩的作用等价为它们对基体结构的集中载荷。已知薄板跨度为板长 l ，则集中荷载为： <math display="block" id="M51"><mi>F</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>M</mi></mrow><mrow><mi>l</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872402&type= 13.88533401 7.53533268 . 如 图5 图5 所示，假设集中力作用于弹性薄板中心处的微小矩形面上。 在弹性力学中矩形薄板弯曲的纳维解基本形式可表示为： <math display="block" id="M52"><mi>w</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub></mrow></mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>x</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>y</mi></mrow><mrow><mi>b</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872421&type= 49.19132996 8.46666718 . 将 式（24） 式（24） 代入薄板弯曲的平衡方程，然后利用积分中值定理得到： <math display="block" id="M53"><mtable><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>l</mi><mi>b</mi><mi>D</mi><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msubsup><mo>∫</mo><mrow><mi>ξ</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow><mrow><mi>ξ</mi><mo>+</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msubsup><mrow><msubsup><mo>∫</mo><mrow><mi>η</mi><mo>-</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>η</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow><mrow><mi>η</mi><mo>+</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>η</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msubsup><mrow><mfrac><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>ξ</mi><mi mathvariant="normal">Δ</mi><mi>η</mi></mrow></mfrac></mrow></mrow><mo>×</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>x</mi></mrow><mrow><mi>l</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>y</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mi mathvariant="normal">d</mi><mi>y</mi><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">4</mn><mi>F</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>l</mi><mi>b</mi><mi>D</mi><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>ξ</mi></mrow><mrow><mi>l</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>η</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872427&type= 69.51132965 41.74066544 将 式（25） 式（25） 代入纳维解基本表达式，得到薄板上任意点（ a，b ）的挠度表达式： <math display="block" id="M54"><mtable><mtr><mtd><mi>w</mi><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">4</mn><mi>F</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>a</mi><mi>b</mi><mi>D</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>ξ</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>η</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>x</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>y</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872432&type= 71.37400055 23.28333473 . 当荷载作用在板中心时， 式（26） 式（26） 可简化为： <math display="block" id="M55"><mi>w</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>F</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>β</mi><mi>l</mi><mi>D</mi></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>m</mi><mi mathvariant="normal">π</mi><mi>x</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>y</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872437&type= 63.33066940 20.57399940 . 3 多目标优化过程及优化结果 1 对上述超声悬浮轴承结构设计的目标函数进行计算，发现增大轴承圆筒的壁厚会导致固有频率增大和圆筒壁挠度减小（即振幅减小），而增大圆筒的长度会导致固有频率减小，但圆筒壁的挠度会增大。即目标之间是相互冲突的，无法使谐振频率和挠度两个优化目标同时达到最优。并且固有频率和挠度的量纲也明显不同，无法直接进行比较，此时传统优化设计方法无法解决该问题。 多目标遗传算法是一种基于遗传进化原理的随机搜索算法，为解决多目标优化问题提供了新的思路 ［ 13 13 - 14 14 13-14 ］ 。本文采用NSGA-Ⅱ多目标遗传算法，引入精英策略，通过计算拥挤度和拥挤度因子保证了种群的多样性 ［ 15 15 ］ 。采用NSGA-Ⅱ对超声悬浮轴承结构进行多目标优化设计，流程如 图6 图6 所示。 3.1 目标函数与设计变量 2 利用前文推导的固有频率与挠度的数学模型作为优化目标函数。在优化过程中，壳体的固有频率和挠度越大越好。建立的多目标优化函数如下： <math display="block" id="M56"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi>w</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi>f</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872451&type= 27.26266861 10.49866676 . 根据目标函数中的共有参数，这里选择圆柱壳体的长 l ，内径 r 和壁厚 h 作为设计变量。根据轴承的加工难度和使用要求，各结构设计参数在一定范围内取值，选取的优化变量范围为： <math display="block" id="M57"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo>≤</mo><mi>l</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>≤</mo><mi>r</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.001</mn><mo>≤</mo><mi>h</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21872458&type= 16.34066582 15.07066536 . 3.2 优化求解 2 本文基于多目标遗传算法编制程序在Matlab中进行求解。设定遗传算法参数：个体总数为100，变异率、交叉率分别设置为0.01和0.8。经过1 000代遗传迭代后得到Pareto解，所有最优解形成帕累托前沿，如 图7 图7 所示。两个坐标轴分别表示两个目标函数，纵轴表示负的固有频率，横轴表示负的挠度。 从Pareto最优解中选取10组最优解列于 表1 表1 中。在这些解中，考虑到实际壳体的加工难度和材料的疲劳强度，壳体结构壁厚不能过小，整体结构的固有频率不能过大。经过筛选，本文选择第6组解作为优化结果。经圆整该优化结果为： l =40 mm， r =10 mm， h =2 mm。 编号 长度/mm 半径/mm 壁厚/mm 1 0.010 302 266 0.005 000 000 0.004 532 538 2 0.015 572 348 0.005 000 000 0.002 000 024 3 0.026 071 723 0.006 600 526 0.002 000 000 4 0.030 244 398 0.009 042 858 0.002 000 000 5 0.036 782 407 0.009 532 030 0.002 000 001 6 0.040 233 704 0.010 428 475 0.002 053 027 7 0.045 300 287 0.013 495 856 0.002 000 001 8 0.051 411 120 0.014 295 314 0.002 021 935 9 0.056 617 090 0.016 960 137 0.002 000 000 10 0.060 000 000 0.024 783 094 0.002 000 000 4 超声悬浮轴承实验 1 为了对本文优化方法的效果进行验证，制作了3组不同尺寸的超声悬浮轴承， 表2 表2 是3组超声悬浮轴承机构尺寸， 图8 图8 是超声悬浮轴承实物照片。 组别 长/mm 直径/mm 壁厚/mm 实验组 40 20 2 对照组A 30 20 5 对照组B 40 30 2 超声悬浮轴承径向振幅测量的实验设备有信号发生器、压电陶瓷控制器和激光位移传感器，测量实验现场如 图9 图9 所示。调节AFG-2225型信号发生器输出正弦交流电压信号并设置频率。信号经E00.D3型压电陶瓷控制器放大一定倍数，频率不变。放大后的电压信号经连接器和导线输送到轴承圆筒外表面的压电陶瓷片，此时轴承在正弦交流信号的驱动下振动。将轴承通过夹持装置固定在隔震测量平台上，调节传感器激光发射源与轴承表面测量点间距到要求的测量距离后，对轴承表面测量点的振幅进行测量，通过信号采集软件读取轴承的径向振动位移数据，如 图10 图10 所示。 在75 V电压下，分别对3组超声悬浮轴承的压电陶瓷片施加3~60 kHz的正弦电压信号，并对每个轴承的振幅进行多次测量，测得的最大振幅随激振频率的变化如 图11 图11 所示。 由 图11 图11 可以看出，实验组振幅在27.21 kHz处出现峰值，峰值振幅为0.38 μm；对照组A振幅在45.32 kHz处出现峰值，峰值振幅为0.18 μm；对照组B振幅在17.10 kHz处出现峰值，峰值振幅为0.42 μm。将优化尺寸轴承的测试结果与理论计算值进行对比，结果如 表3 表3 所示。 理论值 实验值 误差/% 振幅/μm 0.351 0.38 7.63 谐振频率/kHz 30.379 27.21 11.64 由 表3 表3 可知，理论计算值与实验测量结果存在误差，但误差在合理范围内。造成振幅误差的原因主要是环境噪声振动。引起频率误差的原因主要是测量时轴承圆筒的夹持方式、装夹位置和夹紧力不同对轴承的固有频率产生了一定的影响。从实验测试结果可以看出，3组尺寸中振幅最大的是对照组B，谐振频率最大的是对照组A。对照组A虽然振幅较大但其谐振频率较低，对照组B虽然谐振频率较大但是振幅较小。实验组为优化尺寸，在振幅和频率之间取得了平衡，振幅值和频率值均有较满意的结果。 然后，对3组超声悬浮轴承的径向悬浮力进行测量和对比。实验现场如 图12 图12 所示，实验设备有信号发生器、压电陶瓷控制器、光电计时器和精密天平。 假设轴承工作时轴固定不动，此时圆筒壳体内表面和轴的外柱面可近似视为两个挤压面，在高频交流电压的作用下压电片带动轴承圆筒内壁振动，挤压圆筒与轴圆柱面间的空气形成挤压膜。由于样机产生的气膜厚度较小，一般在微米级，很难直接判断轴承是否完全脱离了轴面，因此直接测量轴承最大径向悬浮力存在很大困难。为此设计了测力实验台，通过观察和测量轴承与斜面之间的摩擦状态来间接测量承载力。实验时不断调整倾斜轴与水平面之间的夹角，当轴承对倾斜轴的正压力与悬浮力大小相等方向相反时，即可以近似认为轴承实现了无摩擦悬浮，此时轴承滑下加速度约等于重力加速度。轴承下滑加速度通过光电计时器测出，轴承和负载的质量通过精密天平测出，轴承对倾斜轴的正压力通过轴承和负载的重力、倾斜轴与水平面的夹角简单计算得出。 3组轴承倾斜轴下滑测量径向悬浮力实验结果如 表4 表4 所示。由实验结果可以看出，实验组优化尺寸轴承对气膜的挤压效率最高、形成的径向悬浮力最大，说明前文使用的优化方法是正确、有效的。 组别 倾斜角/（°） 轴承和负载 质量/g 径向悬浮力/N 实验组 39 165 1.28 对照组A 65 106 0.45 对照组B 43 131 0.96 5 结 论 1 本文设计了一种压电片驱动的径向包容式超声悬浮轴承结构，建立了轴承圆筒壳体结构的多目标优化数学模型，并将NSGA-Ⅱ多目标遗传算法应用于结构优化设计中。通过Matlab编程计算，实现了谐振频率和挠度两目标的优化，有效克服了同类型换能器设计时经验依赖性强的缺陷。利用该优化设计方法，获得了基体结构参数的最优组合，即圆筒壳体：长40 mm、直径20 mm、壁厚2 mm。加工制作了3组轴承进行了实验对比测试，实验证明优化尺寸轴承在挠度和频率之间取得了平衡，即振幅值和频率值均有较满意的结果，优化后轴承振幅的理论值为0.351 μm，实验测得的振幅值为0.38 μm，误差为7.63%，谐振频率的理论值为30.379 kHz，实验测得的谐振频率为27.21 kHz，误差为11.64%。通过轴承径向悬浮力测量实验，发现优化尺寸轴承对气膜的挤压效率最高，形成的径向悬浮力最大可达1.28 N，提高了超声悬浮轴承在高频振动下的承载能力，为高效超声悬浮轴承设计奠定了基础。 参考文献 RAI P K ， YADAVA V ， PATEL R K . 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LI Dong-ming,WANG Wan-lei,JIA Ying,et al.Structural optimization of radial containment ultrasonic suspension bearing[J].Optics and Precision Engineering,2021,29(10):2375-2385. 辽宁省教育厅计划项目（No.L2014178）；辽宁省教育厅科学研究项目（No.2018J12GX040）；大连市科技创新基金资助项目（No.2018J12GX040） 2021-06-23 2021-08-10 2021-10-15 2021-11-11 光学精密工程 10 29
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